Kempe chain

Alfred B. Kempe
(http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Kempe.htmlbio at St Andrews)
first used these chains, now called after him,
in 1879 in a “proof” of the four-color conjecture. Although Percy Heawood
found a flaw in his proof 11 years later, the idea of Kempe chains itself is
quite sound. Heawood used it to prove 5 colors suffice for maps on the plane,
and the 1976 proof by Appel, Haken and Koch is also based on Kempe’s ideas.

The original Kempe chains were used in the context of colorings of countries
on a map, or in modern terminology face colorings of a plane graphG
(such that no two adjacent faces receive the same color). The idea was
extended by Heawood to embeddings of a graph in any other surface. Here

•

a Kempe chain of colors a and b is a maximal connected set of faces
that have either of those colors. as in: you can travel from any face in the set to any other, through the set. Maximal as in: there are no more faces of those colors you could enlarge the set with, i.e. that border the area you’ve already got.

In the modern dual formulation of the four-color theorem, faces of G are
replaced by vertices of the dual graphG* (and vice versa); vertices are
adjacent (linked by an edge) in G* whenever the corresponding faces of G
were adjacent (sharing a border). It now becomes a vertex coloring
problem (again, adjacent vertices must receive different colors). Here

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a Kempe chain of colors a and b is a maximal connected subgraph
containing only vertices of those colors. Connected as usual in graph theory
(there’s a path between any two vertices) and maximal as in: there are no
more vertices of those colors you could enlarge the subgraph with, i.e. that
are adjacent to a vertex you’ve already got.

An alternative formulation: let G⁢(a,b) be the subgraph induced by all the
vertices of color a or b (that is, with all the edges that run between
those vertices). Now any connected componentH⁢(a,b) of G⁢(a,b) is a Kempe
chain.

While faces imply an embedding in a surface, the vertex version of the
definition does not rely on any embedding. The chains are more useful in
the context of an embedding though.

Kempe chains of edges

With edge coloring of graphs (again, with different colors for adjacent
edges i.e. those that meet at a vertex), an analogous concept can defined.
Here

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a Kempe chain of colors a and b is a maximal connected subgraph
where the edges have either of those colors. Connected and maximal as before.

An alternative formulation: let G⁢(a,b) be the subgraph consisting of all the
edges of color a or b (with any vertices incident to them). Now any
connected component H⁢(a,b) of G⁢(a,b) is a Kempe chain.

While superficially similar to the previous definition, these “Kempe chains”
are rather different animals. Their structure is far simpler. At any vertex of
H⁢(a,b), there can be at most one edge of color a and one edge of color b
(at most two edges in all). Two things can happen:

∘

Every vertex in the chain has two such edges. In a finite
graph, this must mean a closed path (cycle). Note that, being colored
alternatingly, the number of edges must now be even.

∘

A vertex misses out on having an edge of one of those colors,
and the chain stops there (it can’t miss out on both colors because then
it wouldn’t be part of the chain). In a finite graph, the other end of
the chain must now also terminate somewhere (at a vertex that misses
out either one of the colors). The chain is an open path of one or
more edges (its length can be even or odd).

Kempe chain arguments

One technique that can be used with all these types of chains is

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swapping the colors in one H⁢(a,b). This is always possible:
by definition, there are no adjacent items that could
lead to a color clash.

Kempe slipped up when he swapped an H⁢(a,b) and an H⁢(a,c) without
taking in account that swapping colors a and b in part of the
graph could alter the shape and connectedness of any H⁢(a,c), but
swapping colors in one Kempe chain at a time (and then taking stock of
the lay of the land afresh) is quite safe.

Swapping can be used to free up a color somewhere, see the
http://planetmath.org/node/6932proof of Vizing’s theorem for a repeated use
of this ploy on edge-based Kempe chains.

Specific to the use in plane graphs we have that

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if a Kempe chain forms a cycle, it disconnects the sphere or plane area
into disjoint parts (on the plane, inside and outside the cycle).

Specifically when four colors are used for faces:

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The whole area is divided into red/green swathes and yellow/blue ones.
Alternatively, into red/yellow and green/blue ones. Or red/blue and
yellow/green ones.

Specifically for edge-based Kempe chains in regularρ-valent graphs, if
we try to edge-color the graph with only ρ colors:

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No vertex can miss out on any of the ρ colors, so every H⁢(a,b)
must be a cycle (of even length).